Prescribing the curvature of a Riemannian manifold

=

(3 .32 )

2.:l K )

which, after the change of variables cp = - 2 u (note that A = 0 or A ± 1) is just (1.1 1). If A = 0, then the equation is linear, and has a solution if and only if (3.28) is satisfied. From our work in Chapter I, we see that the case A < 0 should be easier than A > 0, where we anticipate severe difficulties. We seek a solution of (3.31) with g = g0 + cp " positive definite. The existence theorem for (3.31) with A < 0 was first proved by Aubin [Au3]; the first complete proof for A = 0 was given by Y au [Y], who also reproved the A < 0 case. =

THEOREM 3.33. Let (M, g0 ) be a compact Kahler manifold. If A < 0, then given any smooth f there is a unique solution cp of (3.31) with g0 + cp " positive definite. The same is true if A = 0, except that f must satisfy (3.28), and the solution is unique up to an additive constant.

29

RICCI CURVATURE

As an immediate corollary we obtain Kahler-Einstein metrics (for A c 1 ( M) < 0) and the solution of the Calabi problem ( A = 0) :

<

0, i.e.,

CoROLLARY 3.34. Let ( M, g0 ) be a compact Kahler manifold. (a) If c1( M ) < 0, then there is a Kahler-Einstein metric g. Moreover,

if - yg0 is a positive form representing c1(M), then g is the unique Kahler-Einstein metric cohomologous to g0 . (b) Given any closed (1, 1) form representing c1(M), there is a unique Kahler metric g cohomologous to g0 whose Ricic form is One striking application of Corollary 3.34(a) by Yau is to prove that if M is a compact Kahler surface with c1(M) < 0, then 3c 2 (M) � c1(M) 2 , with equality if and only if M is biholomorphically covered by the ball in C 2 • This is proved by writing the characteristic class 3c 2 - c? as an integral involving the curvature of w

w.

the metric. As observed by Guggenheimer [Gu], this integrand simplifies enor­ mously and becomes a sum of squares for a Kahler- Einstein metric, so the inequality and case of equality become obvious. We shall only make a few remarks on how one proves Theorem 3.33 (see [SP or Au4] for an exposition of the details). The existence of a solution of (3.31) can be carried out by the "continuity method". If A = 1 one considers the family of problems -

(3 .35)

det ( g0 + cp") = (det g0 )exp( tf+ cp),

O � t � l.

For t = 0 we have the obvious solution cJ> = 0. Let A be the subset of the interval 0 � t � 1 where one can solve (3.35). A is not empty, and one can prove that it is

open by a routine application of the implicit function theorem. The difficult part is (as usual) proving that A is closed. (Assuming this, then;A = [0, 1] so t = 1 is in A and we are done.) Say t1 E A and t1 � i. To show that i E A we use the solutions cp1 corresponding to t 1, and wish to find a subsequence converging in C 2 (M) to a function -;j,. Then, this limit will solve the equation corresponding to i. By the Arzela-Ascoli lemma, it is enough to find a uniform estimate on l et>) in the + Holder space C 2 a for some 0 < a < 1 . This is done by proving an a priori bound on every solution of (3.35) (3 .36)

l ct> lc2+•

�

const,

where the constant is independent of t E [0 , 1]. The uniform estimate lcJ> I co is an easy consequence of the elementary maximum principle, while the higher derivative estimates are more complicated. One first uses a special argument to estimate l dcJ> I, which then implies that (3.35) is uniformly elliptic. Assuming this, the original proof used a difficult procedure devised by Calabi to estimate lcJ> I c'• which of course gives (3.36). Recently Evans (see [GT, 2nd ed., Theorem 17.14]) has given a rather general proof of interior C 2 + a estimates for uniformly elliptic equations; since we already know the uniform ellipticity and, since on a compact manifold every point is an interior point, Evans' result applies to give (3.36) and completes the proof.

30

RICCI CURVATURE

The existence proof for (3.31) if A = 0 uses the continuity method applied to det ( g 0 + <J>") = ( det g0 ) ( 1 + t(ef - 1) ) , 0 .;;; t � 1 , which one writes symbolically as F( <j> ) = t ( ef - 1), and normalizes <j> by f <j> dV0 0. For this case A 0, all the steps for A < 0 go through without change, except that here the maximum principle does not apply, so one needs to work harder to obtain the uniform estimate l l co � const. Yau's original estimate was simplified in [K2] for complex dimension one and this simplification was ex­ tended to all dimensions in [Au6]. The idea of this simplification is to estimate ll u for all p and let p � oo . Theorem 3.33 does not treat the case A > 0. In this case, just as in the one complex dimensional version (3.32) considered in Chapter I, there are no ade­ quate existence theorems. Until recently, all that was known was an indirect proof of nonexistence of a Kahler-Einstein metric on some manifolds with c1 > 0; this was implied by Matsushima's theorem which asserts that in this case the complex Lie algebra of all holomorphic vector fields on M must be reductive. For nonexistence one anticipated there should be an extension of the Kazdan-Warner obstruction (1.1 2) to higher dimensions. This was found by Futaki [Fl] (see also [C2 and FM]) who proved that if the compact Kahler manifold (M, g0 ) (with g 0 =

=

representing c1(M) > 0) has an Einstein metric-or even a metric with constant scalar curvature-then for all Kahler metrics g cohomologous to g0 ( 3 .37)

where X is any holomorphic vector field and I is a solution of D.f = sg - sg (here s is the average of S as in (3.15)). To relate this to the obstruction (1 .1 2) for the standard sphere ( S2, g0), one begins with a Kahler-Einstein manifold (M, g0 ) with c1 > 0, so Pg0 = Ygo · Given a real function j, seek a Kahler metric g cohomologous to g0 , with Pg - Yg = i'dajj2rr. As before, from (3.26), this leads one to solve det ( g0 + ") = ( det g0 ) e-U+ .P l . ( 3 .38) Since dVg = exp[ - (/+ <j> )] dVgo ' (3.37) then reads

f X( / ) e -U+.P) dV

( 3 .39)

M

go

=

0.

If we let <j> = - 2 u and K = e- f, then (3.38) becomes (1.11) and (3.39) is the old obstruction (1.12). (Note that the space of solutions tf; of D. K tf; + tf; = 0, i.e. D.tf; + 21/; 0, is isomorphic to the space of holomorphic vector fields X by the correspondence tf; - \71/; (see [Ll]). On ( S2, g0 ) the solutions of D.tf; + 2 1/; = 0 are the first-order spherical harmonics.) So far, there are no existence proofs of Kahler-Einstein metrics in the case c1 > 0 on a compact Kahler manifold (M, g) with complex dimension m. Aubin [AuS] has conjectured that if =

J

M

c i" < (m + 1 )2m ( 2 m) - m ,

31

RICCI CURVATURE

then there does exist a Kahler-Einstein metric. He has reduced the proof to the following as yet unproved inequality: There are constants b and c so that for all a � 1 and all cp E C00( M) with g + cp" positive definite

where

I( cp ) =

m2 m j

M

cp

[

1

-

det ( g + cp") det g

] dV.

In the non compact case, Mok and Yau [MY] have some results concerning the existence of complete Kahler-Einstein metrics. (c) Another variational problem. Calabi [C2, C3] has proposed investigating another variational problem on a compact Kahler manifold (M, g0). Find critical points of the functional ( 3 .40 )

among all Kahler metrics g cohomologous to g0• He has proved : (i) all critical points are local minima; (ii) if there is such a metric g with constant scalar curvature, then g gives the absolute minimum of J and every critical point has constant scalar curvature with the same value of J; (iii) a critical metric g has constant scalar curvature if and only if the Futaki obstruction (3.37) is satisfied. On the negative side, Levine [Le] has given examples of Kahler manifolds- they are even projective algebraic varieties-where J has no critical points. In his examples, the connected group of holomorphic transformations is a nonabelian, simply connected, nilpotent Lie group. This contradicts some properties proved earlier by Calabi. It would be good to find a more direct obstruction to the existence of a critical point, since it should be useful in determining just when J does have critical points.

IV. Boundary Value Problems

1. Surfaces with constant mean curvature. Minimal surfaces are the simplest examples of surfaces with constant mean curvature; their mean curvature is zero. To be led to surfaces with constant nonzero mean curvature, one may seek a surface of least area enclosing a region whose volume is a given constant. Spheres are the obvious examples, the sphere of radius R having mean curvature 11 R. It is conjectured that round spheres are the only compact surfaces immersed in R3 having constant mean curvature; in more picturesque language, " the only soap bubbles are spheres". In 1958 A. D. Alexandrov gave a beautiful proof (see [ Sk] ), using only the maximum principle, that this is true if the surface is assumed to be embedded in R3 (see [R] for a different proof). In higher dimensions there is an analogous problem. Alexandrov's result holds in all dimensions so there are no embedded hypersurfaces except round spheres. Furthermore, H. Hopf has proved that there are no immersed topological spheres in R3 with constant mean curvature except for the round sphere. Recently, W.-Y. Hsiang, Teng, and Yu immersed as [HTY] found other examples of compact nonround spheres " surfaces of revolution" in R4 with constant mean curvature. Wente [We4] has just announced that he has found an immersed torus T2 in R3 with constant mean curvature. The possibility of similar examples for surfaces of higher genus is unknown.

S2

'

Rellich problem. s

S3

As every child knows, one can obtain soap bubbles with boundary, i.e. surfaces of constant mean curvature, by taking a soap film spanning a curve and creating a pressure differential by blowing on one side of the soap film; the pressure and the boundary curve then determine the mean curvature. Given a closed curve f in R , we may seek a surface M having r as its boundary and having prescribed mean curvature H. The classical Plateau prob­ lem, which seeks a minimal surface (so H = 0), is an example. This vague formulation does not at all specify the topological type of M. It has long been known that, for instance, some curves are boundaries of both orientable and nonorientable surfaces (see Figure A). Thus we must be more specific.

3

33

34

BOUNDARY VALUE PROBLEMS

FIGURE A Let 0 be the open unit disc in R2 , 0 = { x 2 + y 2 < 1 } and y: aO ---+ R3 a smooth nonconstant Jordan curve, y(aO) = r. Given a smooth function H( x, y), we seek a function u: Q ---+ R 3 having mean curvature H and agreeing with y on ao. The partial differential equation is (using the standard "cross product" of vectors in R3 )

( 4 .1 ) with either Dirichlet boundary conditions u = y on a o , ( 4 . 2 )( d ) o r Plateau boundary conditions

( 4 . 2 )( p )

l uxl 2 = l u l 2

and

v

u ( ao) = r

and

ux . uy = 0

on a o , with

u " nondecreasing"

on ao.

One obtains equation (4.1) as follows. Using isothermal coordinates, write the first and second fundamental forms of the immersion u: 0 � R3 as II = e dx 2 + 2f dx dy + g dy 2 • The standard formula for the mean curvature H is H

+ g) = N · D.u = ( e 2E 2E '

where N = ux X uvfE is the unit normal vector. (Note, E = lux X uvl · ) Since the coordinates are isothermal, then luxl 2 = luvl 2 = E, while ux · uv = 0. Differenti­ ating these equations, one finds that D.u = uxx + uvv is orthogonal to the tangent vectors ux and U y - Hence, D.u = t( x, y) N for some scalar function r. Using the formula above for H, yields (4.1). Note that this equation is valid only if x and y are isothermal coordinates. If r is a circle of radius c, and H is a constant with He < 1, then there are two spherical caps (a large and a small one) of radius 1/H that span r and have mean curvature H, while if He > 1 it is geometrically plausible-and was proved by Heinz [He]-that there are no such surfaces (see Figure B). From this example, Rellich conjectured that for H sufficiently small there are always at least two solutions. Surprisingly, it is unknown, even in the simplest example shown, if there can be more than two solutions.

35

BOUNDARY VALUE PROBLEMS

FIGURE B In 1 970 Hildebrandt [Hi] proved that there is always at least one solution, while Brezis and Coron [BC] proved the existence of at least two solutions with constant H if r is in a ball of radius c and He < 1 (independently, Struwe also proved the existence of at least two solutions for all " sufficiently small" H). We should point out a related problem. Stated intuitively, one seeks a surface M spanning a curve r and containing a fixed volume V (that is, fix any surface M0 that spans f; then V is the volume " between" M and M0 ). This gives a surface M with constant mean curvature whose value is determined by V. From a physical viewpoint, this problem is more natural, while for geometry, it is more natural to fix the mean curvature, not the volume. Basic work on this more physical formulation was done by Wente [Wet, We2], whose contributions are also important in resolving the geometric formulation. We discuss only Dirichlet boundary conditions (4.2)(d). Hildebrandt obtains his solution by seeking a minimum of the functional

E(v)

=

j l vv l2 dA + j v · ( vx £!

£!

X

vY ) dA

for v in an appropriate space of functions satisfying the boundary condition (4.2)(d) (this argument is repeated in [BC]). Using this " small"solution w, Brezis-Coron seek a second solution of (4.1)-(4.2)(d) in the form u = w + v, and obtain v by minimizing the functional

(4 . 3 ) with v in the Sobolev space HJ(Q) and satisfying the constraint

( 4 . 4) Note that since Q ( v) = 1 , then v 'iE 0 and so u = w + v is different from the first solution w. It is interesting that the variational problem (4.3)-(4.4) has many of the same difficulties found in the variational problem (2.6)(a), (b) for scalar curvature. In fact, Theorem 2.ll(b) served as a strong influence on this proof. After proving

36

BOUNDARY VALUE PROBLEMS

that J( v) is bounded from below for v satisfying the constraint Q( v) = 1, let a inf J( v ) for Q( v) = 1 , so there is an HJ weakly convergent minimizing sequence vi --') v, J( v ! a. The trouble is, as before in (2.6)(a), (b), that the weak convergence is not strong enough to enable us to conclude that the limiting function v also satisfies the constraint Q( v) = 1 . Let 2 S = inf vv ,

)

=

u E Hfi ( O ) Q( u ) = l

jl l 0

so 1 /S may be thought of as the best constant in the inequality IQ(v)l213 � S - I_[ 1 \7 v 12• It turns out that this infimum is not achieved unless Q = R" (see [We3]), in which case this constant is achieved by the family of vector-valued functions ..J..' ( ( x , y, e e > 0, x, y) = , '�" (4 .5)

)

x2 + y2 +

e

and their translates. The key inequality that makes the argument work i s the strict inequality a = inf J v < S. v e H!J ( O ) (4.6)

( )

Q( u ) = 1

This is analogous to the situation in Theorem 2.11(b). To prove (4.6), by a translation we may place the origin at a point where \7 w if:. 0, and use the extremal functions (4.5), cut off outside Q, v' = fcp', where f E q'"(U) with f = 1 near the origin. Then by a computation using (4.5) and (4.6), as e --') 0 we find the asymptotic expansion J v ' = S + SH wx O i + wy O · e + o e . Now one chooses a basis � ], k for R3 so that the coefficient of e is negative, thus proving (4.6). The remainder of the proof is quite similar to Theorem 2.ll(b ). One has a minimizing sequence vi --') v of J( v ) with Q( v = 1, where vi converges weakly to v in HJ ( U ). One shows that

( )

[ ()

( ) J]

·

()

)

J I ( vi - v ) 12 dA i j I �

\7

\7 ( v1

) 2 dA +

-v 1

o

(1 ) .

)

Since a < S this implies that vi --') v strongly in H{j. Hence 1 = Q( v = Q ( v ) and the proof that J achieves its minimum is complete. Finally, it remains to show that u = w - ( aj2 H) is actually another solution of (4.1)-(4.2)(d) (see [BC] for more details). 2. Some other boundary value problems.

)

(a) Graphs with prescribed mean curvature. If u = u ( x 1 , . . . , x , is a graph in R" + 1 , then its mean curvature H is (4.7)

. dIV

( VI + 1 V ul2 ) V' U

= nH .

37

BOUNDARY VALUE PROBLEMS

Thus H is the average of the principal curvatures H = ( k 1 + + k11)/n , so for a sphere of radius R one has H = 1/R (note that some mathematicians write H kl + + k11). On a bounded open set n in R" with smooth boundary, given a function H one may seek a solution of (4.7) in n that satisfies the Dirichlet boundary conditions u cp on an. (4.8) If the curvature is too large relative to n, then a solution of (4.7) may not exist. To see this analytically, let E c n be any smoothly bounded open set. Then integrating ( 4. 7) over E and using the divergence theorem we find ·

=

·

·

·

·

·

=

(4 .9)

where v is the unit outer normal on a£ and l aE I is the measure of aE. Moreover, if we require that "V U be bounded in n, that is, u E C 2 ( n ) n C 0• 1 ( 0), then the stronger inequality (4 . 10)

must hold for some constant c < 1 for all E c n. There is a fairly sharp existence theorem covering the Dirichlet problem, part (a) below being primarily due to Serrin [Se], and part (b) to Giusti [Gi]. THEOREM 4.1 1 . Let n be a bounded domain in R" with smooth boundary and let H be a smooth function on n. (a) There exists a ( unique) solution u E C 2 ( n ) n C 0• 1 (0) of (4.7)-(4.8) for arbitrary cp E C 2( 0 ) if and only if (4.10) holds and nH( x ) � ( n - 1)H11 _ 1( x ) for all X E a n , where H - 1 is the n - 1 dimensional mean curvature of an. n (b) There exists a solution of (4.7) in n if and only if (4.9) holds. Moreover, in the extremal case when (4.10) does not hold, the solution is unique up to a constant and the solution is " vertical " on an. As an example of part (b), let n be a disc of radius R and H = 1jR , so the unique solution- up to constants-is a hemisphere of radius R . In part (a), if the boundary restriction on the curvature does not hold, then there is some boundary value cp for which there is no solution. For the special case of a minimal surface in R3 over a set n, the condition in part (a) states that n must be convex. One can ask similar questions in the case of higher codimension, and then run into many fascinating situations with new, almost unexplored phenomena. Fol­ lowing [LO], let n c R " be an open set and consider a C 2 minimal immersion F: n � R " + k . The analogue of a graph is when F has the " nonparametric" form F( x ) (x, u ( x )) for some vector-valued function u: n --+ Rk. Then the " minimal surface" equation is the system =

(4 . 1 2)

38

BOUNDARY VALUE PROBLEMS

where g iJ is the inverse of the metric gij = 8 ij + ( a u;ax i a u;ax i ) induced on the surface (we have assumed u E C 2 to get (4.12)). For the Dirichlet problem one is given a function cp: ao � Rk and seeks a solution satisfying (4.12) and the boundary condition '

u = cp on a o .

(4 . 1 3 )

In Theorem 4.11, for k = 1 we have already noted that 0 must b e convex. For higher codimension, that is k � 2, Lawson and Osserman found that new complications arise. We give only one of their results. Let D n be the closed unit disc in Rn and aD = s n - 1 . Note that D n is convex. Then the homotopy class of the boundary value cp is important to the solvability of (4.12)-(4.1 3). To be more specific: THEOREM 4.14 [LO]. Let c/Jo : s n - 1 � s n - 2 � Rn - 1 be any C 2 map not homo­ topic to zero as a map into s n - 2. There is a constant A 0 > 0 so that if A � A0, then there is no solution to (4.12) in D n with u = Acp on ao ( while there is a solution with u = Acp on ao for all A near zero). Existence for small A follows from the observation that if A = 0, then u 0 is obviously a solution, and one can use the implicit function theorem to obtain a solution for all A near zero. The simplest example where the nonexistence arises is when cp0: S 3 � S 2 is the Hopf map. In this case n 4 and the codimension k = 3, the original domain is just the standard ball D4 and the boundary value cp0 is a quadratic polynomial =

=

c/Jo

=

( l z l - l z l , 2 z 1 z2 ) ,

where z 1 x 1 + ix , z 2 = x 3 + ix 4 , S 3 is thought of as the unit sphere in 2 C 2 = R4 , and S 2 as the unit sphere in R X C = R 3 • Clearly this example does not indicate perverse pathology, but an honest obstruction. The proof of Theorem 4.14 in [LO] uses special properties of minimal surfaces. So far, ther is no similar nonexistence assertion for othe quasi-linear systems, such as the system for prescribed mean curvature. Perhaps an analogous situation is the quasi-linear system for homotopy classes of harmonic maps (see [ELl and EL2, Il.1] for a survey and references). The situation clearly needs to be understood more fully. =

(b) Graphs with prescribed Gaussian curvature . The Gaussian (or Gauss-Kronecker) curvature K of a hypersurface in Rn + 1 is the product of the principal curvatures. For a graph u = u(x 1 , . . . , x ) the formula for K is n ) ( + )/ det( u ;1 ) = K ( x ) ( 1 + l v u l 2 n 2 2 , (4 .15) where u ;1 = a 2 ujax; ax1 is the hessian of u. Some aspects of the n = 2 dimen­ sional case were discussed in Chapter I. In higher dimensions, essentially nothing is known about (4.15) as a partial differential equation for u, given K, except when K > 0 so the equation is elliptic. For the case K > 0, attention has

39

BOUNDARY VALUE PROBLEMS

primarily been given to he Dirichlet problem, where one is given a bounded strictly convex domain n with smooth boundary and a function cp on an. The problem asks for a convex function u with (4.16)

u

= cp

on a n .

One can easily obtain a necessary condition for the solvability of (4.15), independent of any boundary conditions, by making the change of variables y \7U(x) : n -+ Rn =

( 4.17)

1 J 1Q K dx = 1Q ( 1 + i "V u i z ) ( n + 2)/2 dx � R" ( 1 det u i

+

dy IYI

2 ) ( n + 2)/ 2

=

wn _ 1

--

where W _ 1 is the volume of the unit Sphere s n - 1 . If one requires that n bounded in n , then strict inequality must hold in (4.17), that is, (4 . 1 7 ' )

1 K dx o

<

n

'

\7U

is

wn - 1 . n

Trudinger and Urbas [TU] have proved the analogue of Theorem 4.ll{a), and Urbas [U] found the result corresponding to Theorem 4.ll(b), which treats the case of equality in (4.17). Until recently, the existence proofs for real Monge-Ampere equations such as (4.15) with n ;;. 3 first found approximating solutions by convex polygons, and then passed to the limit (see [Pl, P2]). Cheng and Yau [CY] and P. L. Lions [Lio l , Lio2] gave more direct existence proofs. But until the past year or so there were no proofs based on a priori estimates, as needed for using the continuity method, Schauder's fixed point theorem, or Leray-Schauder degree theory. The difficulty was obtaining a priori c 2 + a( O ) estimates on the solution, assuming one already has a uniform estimate for second derivatives of u. Caffarelli, Nirenberg, and Spruck [CNS] and Krylov [Kr2] have independently proved the desired inequality. For the interior c 2 + a(n) estimate, one can often use the result of Evans (see [GT, Theorem 17. 14]) or of Krylov [Krl], so one just needs the boundary c z + a estimate. W e shall discuss this in the next section. 3. The C 2 + a estimate at the boundary. We consider solutions of the nonlinear boundary value problem

(4.18)

F( x , u, Du , D 2 u )

=0

in n

with Dirichlet boundary conditions (4.1 9)

u

= cp

on a n .

Here n -+ Rn i s a bounded domain with smooth boundary, F(x, u, V ; , riJ ) and cp are assumed to be smooth scalar-valued functions of their arguments, and (4.18) is assumed elliptic in the sense that the matrix (aF;ariJ ) is positive definite for appropriate values of the arguments. For the Monge-Ampere equation (4.15), this means u is required to be convex and K > 0. In this case, n is also assumed to be strictly convex.

40

BOUNDARY VALUE PROBLEMS

Three traditionally useful methods of proving the existence of a solution (the continuity method, Schauder's fixed point theorem, and Leray-Schauder degree) all require finding a priori estimates (in a space such as the Holder space C 2 +"(Q)) of any solution of (4.18)-(4.19). For many problems, it was known how to obtain uniform estimates for the second derivatives; these estimates depend on special properties of the equation. Even though these were known, until the past year or so it was not known how to obtain the additional estimates in C 2 +"(Q). Caffarelli, Nirenberg, and Sprock [CNS] (see also the exposition in Chapter 17 of [GT]) and Krylov [Krl, Kr2] have independently shown how to prove the desired estimates at the boundary. Even more recently, Caffarelli has simplified Krylov's basis estimate. We shall present the details of Krylov's boundary estimate after first outlining how, presuming the other estimates have been carried out, one can use this estimate to find an a priori C 2 +"(Q) bound on solutions of (4.18). Let u E C3(Q) be a solution of (4.18)-(4.19). Replacing u by u - cp, there is no loss of generality if we assume cp = 0. Suppose that there are constants K1 and K 2 so that any such solution satisfies (4 .20)

(where we let estimate

lul 2 = sup0 lui + sup0 iDui + sup0 ID 2 ul) ,

as well as the interior

(4.21)

for any n 0 <E n. Also, assume that for these solutions the linearization of (4.18) is uniformly elliptic, that is, for some constants 0 < ;\ < A the matrix (aF;ariJ ) is uniformly positive definite, (4 .22)

0

To obtain the required C 2 +" estimate of u near a boundary point x , we first 0 flatten the boundary near x , that is, by a diffeomorphism we may assume that 0 0 near x the boundary of n is the hyperplane x n = 0 with x at the origin (see Figure C). Use coordinates where x n > 0 corresponds to the interior of n. We want to estimate the second derivatives u i J on an. Since cp = 0, then u = 0 on an so the tangential derivatives

r FIGURE C

>

X

I

41

BOUNDARY VALUE PROBLEMS

of u on the boundary are all zero, u i1 = 0, 1 .::; i, j .::; n - 1 . Thus we need only estimate the normal derivatives u1 n , u 2 n , . . . , u nn on an. But by the ellipticity of (4. 1 8), we know aFjarnn > 0. Thus the equation (4.18), F = 0, can be solved for

u ,,:

1 .::; i , j .::;

n -

1,

i n terms o f the other derivatives. Consequently, we need only estimate the mixed normal derivatives u k n for 1 .::; k .::; n - 1 . After we estimate these u k n in the c a-norm on an, then the desired estimate (4.23) l u l c z + • (fl ) .::; K3 follows. For details of how one uses (4.23) to prove the existence of a solution, we refer to the discussion of (3.35) above, to [CNS, §§1 and 7] and to [GT, Chapter 17]. Our task is to estimate the mixed normal derivatives u k n for 1 .::; k .::; n - 1 in the c a-norm on an. Take the derivative of (4.1 8) with respect to x k and obtain n

aF

u i , � l aril ilk

(4 .24)

=

h (x ) ,

where h ( x ) involves the other terms from computing a;ax k of (4.1 8). From our assumption (4.20) we already know that h(x) is a bounded function in 0. Let w = u k and let a ii = aF;ari1. We want to estimate aw ;ax n in ca. Equation (4.24) is of the form

i ,l

w= 0

on a n .

Earlier assumptions imply that (a ii ) is uniformly positive definite in n,

0 < ;\ .::; ( aii ) .::; A , (4.25) that w E C 2( 0 ) with lwl , I Y'wl , and l h l bounded in 0 . Note that ail will usually depend on the second derivatives u J of u. Thus, since we do not (yet) have a i uniform Holder continuity estimate on the u J • we cannot make any uniform i Holder continuity assumption on the ail. To summarize, we let w = u k and our goal is to estimate wn = u k n in the c a-norm on an (in [CNS] they let w = u n and estimate wk in the ca-norm on

an).

It is sufficient to consider the following linear problem: In a half-ball B + c of radius four (say), we have a solution u E C 2 ( ii +) of

( 4 .26)

n

Lu := - L a ii ( x ) u i1 = J(x) i,j= l

in B + ,

Rn

u (x', O) = 0 ,

where (ail) and f are continuous (bounded measurable i s enough), (4.25) in B + , and (4 .27) l u l , l v u l , lf1 < K in B + for some constant K (see Figure D). Let r denote the disc hyperplane x n = 0.

(aii ) satisfies

lx'l .::; 1

in the

42

BOUNDARY VALUE PROBLEMS

-1

x'

FIGURE D THEOREM 4.28 (KRYLOV [Kr2]). Let u E C 2 ( Ji+) be a solution of (4.26), and assume that (4.25) and (4.27) are satisfied. Then there are constants a , 0 < a < 1 , and c depending only on n , k, \ , and A such that

lo ujox nlc � c.

(4 .29)

PROOF. Since u = 0 on r,

au

o x n (x') =

lim

x. -- o

u ( x', x n ) - 0 xn - 0 =

lim

x. -- o

u(x)

xn

;

thus we estimate

v ( x ) := u (x )/x n .

(4.30)

It is convenient to introduce some notation. For R can take ll = \ j( 9n A ) < 1/2), let

� 1 and some fixed ll > 0 (one

Q ( R ) = { l x' l � R , O � x n � llR } , Q + ( R ) { l x' l � R , ll R/2 � x n � llR } , =

and m

R

=

inf

Q( R )

v,

MR = sup

(see Figure E). We also write sup l f l = sup8 + l f l

Q( R )

v

� K.

Q + (R)

�- - ,

Q(R /2 )

1

R/2 FIGURE E

oR R

X

I

43

BOUNDARY VALUE PROBLEMS

Two preliminary results are needed. The first incorporates Caffarelli's simplifi­ cation of Krylov's original proof-which treated the above function v rather than using u directly as we do below. LEMMA 4 . 31 . If Lu � f in Q( R ) with u � inf v

(4.32)

Q\ R )

�

<

inf v +

Q ( R /2 )

U

where v is defined in (4 . 30).

0 and u(x', 0) = 0, then

� sup l/ 1 , 1\

PROOF. Let y : inf v ( x ) for (4 .33) and introduce the comparison function =

( 4 .34 )

Z

( X ) .· -

YX n

(tu

2u!:'

x n = 8R , xn

lx'l 2

)

lx 'l < R ,

x n ( 8R - x n )sup l/1

. + R 2A R2 It is straightforward to verify that if 0 < 8 < 1/2, then in Q ( R ) (i) z( x', 0) = 0, (ii) z(x) < 0 on the sides of Q ( R ) : { lx'l = R, 0 < x n < 8R } , (iii) z(x) < 2y8 2R < y8R on the top of Q ( R ) : { lx'l < R , x n = 8R } . Also (iv) Lz < - sup l / 1 < f if 8 > 0 is sufficiently small. In particular, these all hold if 8 = Aj9n A . Since u � 0 i n Q ( R ) and by (4.33) u = x n v � y8R on the top o f Q ( R ), then L ( u - z) � 0 in Q ( R ) with u � z on
-

(

_

�

)

The second lemma we need is a Harnack-type inequality. Let the ball centered at x and having radius r.

B(x;

r) denote

LEMMA 4.35. Let u E C 2 ( li +) satisfy Lu = f in B + with u � 0 in B(x; 4r ) Then there is a constant c > 0 depending only on n , A , Q, and K such that (4.36)

sup u

B(x; r)

<

c

(

sup

Q+( R )

v

<

c

(

B +.

)

inf u + R 2 sup l f l .

B ( x ; r)

Consequently, if Lu = f in B + with u � 0 in Q(2 R ) ( with R new constant c > 0 depending only on n , A , Q and K so that (4 .37)

c

<

1), then there is a

)

inf v + R sup l/ 1 .

Q +( R )

PROOF. Inequality (4.36) is an immediate consequence of Theorems 9.20 and 9.22 in [GT] . (To find the factor R 2 in (4.36), first prove (4.36) for R = 1. Then making the transformation x --+ xjR changes Lu = f to Lu = R 2f.)

44

BOUNDARY VALUE PROBLEMS

To prove B(x, 8Rj2) 4r 8Rj2. x E Q +( R ) , =

(4.37), observe that about every point x in Q +( R ) there is a ball in Q +(2R ). Because u � 0 in Q (2 R ) we can apply (4.36) with Since a finite number (independent of R ) of the balls B(x, 8Rj8), cover Q +( R ), we conclude that, with a new constant c,

u < c(

sup

(4.38)

Q +( R )

But in Q \ R ) we have 8Rj2 1 2 8R sup

Q +( R )

inf

Q+( R )

u + R 2 sup lf l ) .

� x n � 8R. Since u = x nv this gives

v�

sup

Q +( R )

u

and

inf

Q +( R )

u � 8R

inf

Q+( R )

v.

Combined with (4.38) this proves (4.37) with some possibly larger constant Q.E.D.

c.

We now complete the proof of Theorem 4.28. In what follows, we assume R � 1 and c 1 , c 2 , etc. denote constants depending only on K, n , A , and A. By Lemma 4.35, in particular (4.37), with u replaced by u - m 2 R x n � 0 in Q(2R ), we obtain sup

Q+( R )

( v - m 2 R ) � c1 [

inf

Q +( R )

( v - m 2 R ) + R sup lf l ] ·

Thus, using Lemma 4.31

� c2 [

inf ( v

Q( R /2 )

- m 2 R ) + R sup lf l ]

c2 [ m R 12 - m 2 R + R sup If I ) . . these same inequalities with u replaced by M2 R x n - u � 0 in =

Repeating w e find that

sup

Q +( R )

Q(2 R )

( M2 R - v ) � c 2 [ M2 R - MR12 + R sup lf l ) .

Adding these last two inequalities thus gives

M2 R - m 2 R � c3 [ ( M2 R - m 2 R ) - ( MR12 - m R12 ) + R sup lf l ) . Let w ( R ) = MR - m R denote the oscillation of v in Q ( R ). Then inequality (4.39) (4 . 39)

states

(4 .40)

w ( R/2)

� Ow (2R ) + R sup lf l ,

where 0 < (} = 1 - 1jc3 < 1 . Making (} larger if necessary, we may assume that 1 /4 < (} < 1 . Repeatedly using (4.40), a straightforward inductive argument gives

m = 1 , 2, . . . . Hence (4 .41 )

BOUNDARY VALUE PROBLEMS

45

with 2 sup l / 1 . 40 - 1 Note that (4.27) shows that w (l) is bounded. Given any R � 1, there is an integer m so that 1j4m < R � 1/4m - l . Thus m log(l/4) < log R and hence o m < Ra, where 0 < a : = log 0 /log(l/4) < 1. Combined with (4.41) this yields

c

w(R) Letting x

n

--+

�

4

w

=

w (l)

( 4 �- 1 )

+

�

c4o m- 1

0 in this we find that in the set f

<

�4 Ra .

Q(R) with R � 1 , oscillation of ( o ujoxn ) � c5 R a . n

This is the desired Holder estimate (4.29). Q.E.D.

Some Open Problems

The following is a summary of some of the open problems mentioned in these lecture notes. Most of these are well known.

1 . Given a smooth function K(x, y), find a graph z = u ( x , y ) in R3 whose Gaussian curvature is K(x, y ) in some neighborhood of the origin. Thus, one wishes to solve

locally near the origin. The cases K(O) > 0 and K(O) < 0 can be solved by standard elliptic and hyperbolic machinery, while the case K(x, y) > 0 and the case K(O) = 0, vK(O) =I= 0 have both been solved recently by Lin. The general case where K(O) = 0 is unresolved- there may possibly even be situations where one can prove that no solution exists locally. 2. On a compact two-dimensional Riemannian manifold one knows that the equation d u = h ( x ) e u ( h =I= 0) has a solution if and only if both f h > 0 and h change sign [KWl]. Th�.:se necessary conditions must still be satisfied in the n > 3 dimensional case too, but no sufficient conditions are known. The exponential e u causes difficulty if one uses the calculus of variations. This is probably the simplest differential equation about which we know so little. 3. On S 2 with the standard metric, for which functions l( x ) can one solve du = 1 - l( x ) e 2u? There are some necessary conditions [KWl, BE], but no adequate sufficient conditions (except that if l( x ) = I( - x) and I is positive somewhere, then it is known that a solution does exist [M]). As a test, an adequate understanding should enable one to prove the known fact (by the uniformization theorem) that ( S 2, g ) has a pointwise conformal metric with· curvature 1. To start, it may be useful to consider the special case where I is rotationally symmetric and seek a solution of the related ordinary differential equation. 4. Except for some results on Rn , there is little known about pointwise conformal deformation of metrics on complete open manifolds. The simplest case which should yield new phenomena is if M n is the Poincare disk Hn with 47

48

SOME OPEN PROBLEMS

curvature - 1 . Which functions K( x ) are Gaussian (or scalar) curvature functions of complete pointwise conformal metrics? This involves solving tl. u - 1 - K( x ) e 2 u on H 2 , =

- 4( n - 1 ) tl. u + S u S ( x ) u < n + Z)/( n - Z) on H " n >- 3 ' """ ' ° n -2 where S0 = - n ( n - 1) is the scalar curvature of H " . New phenomena are expected since on H" one can uniquely solve the Dirichlet problem for tl. u = 0 with continuous boundary values, whereas on R" the only bounded harmonic functions are constants. Bland and Kalka [BK] have some results. See also [AvM]. One should also consider related boundary value problems on compact mani­ folds with boundary. Cherrier [Ch] has some results. =

5. Recently there have been some results (by Oliker, and Treibergs and Wei) on graphs over S 2 with prescribed Gaussian or mean curvature. The next simplest case may be graphs over the torus, T 2 � R3, but there are many open questions here.

6. Harmonic spinors and the Dirac operator have been important in finding topological obstructions to compact (or complete) manifolds having metrics with positive scalar curvature (see [GL1-GL3]). Is there some way to use harmonic spinors to construct metrics with positive scalar curvature? There is a similar question concerning harmonic 1-forms and positive Ricci curvature. 7. Does every noncompact manifold have a complete metric with constant negative scalar curvature? This is known to be true for compact manifolds. 8. Solve the analogue of the Yamabe problem for complete noncompact manifolds, as well as some version on compact manifolds with boundary. 9. Does every compact M", n � 3, admit a metric with negative Ricci curva­ ture? Gao and Yau have announced that there is always a negative Ricci curvature metric on any compact M3. What about the analogous question for n � 4 and for complete metrics on noncompact M " ?

10. The only known obstructions to Einstein metrics on compact manifolds M " are when n .;;; 4 . Are there any in higher dimensions? One expects the answer is " yes". 1 1 . Which symmetric tensors R, 1 are locally the Ricci tensor of a metric? DeTurck [D2] proved that R,1 is locally a Ricci tensor if R;1 is invertible, but found some obstructions if R;1 is not invertible. The natural guess is that R;1 is locally a Ricci tensor if there is some metric g so that the second Bianchi identity is satisfied locally. This is unresolved even in the real analytic case. 12. If a compact M" has a metric with positive Ricci curvature, can it be deformed to an Einstein metric with positive Ricci curvature? Hamilton's theorem [Hal] shows this is true if n = 3.

SOME OPEN PROBLEMS

49

1 3 . If a compact M 3 satisfies the assumptions of the Poincare conjecture, does it admit a metric with positive Ricci curvature? If so, then by Hamilton's theorem [Hal] the Poincare conjecture is resolved. 14. Which compact Kahler manifolds with the Chern class c1 > 0 have Ein­ stein-Kahler metrics? Is Futaki's obstruction [Fl] the only one? This is closely related to problem 3 above. 1 5 . Calabi has asked when, on a compact Kahler manifold m with metric g0, there is always a cohomologous metric g = g0 + zz minimizing the square of the scalar curvature J( g) = f S 2( g ) dVg. For example, if c1( M) > 0 and does not admit a Kahler-Einstein metric, then such a metric minimizing J is a candidate for a " nice" metric on M. Levine [Le] has examples where no minimum exists. When does the minimum exist? 16. Minimal surfaces have recently shown their value as an analogue of geodesics in proving results in geometry. Is there a fruitful generalization of such ideas as " geodesic flow", "comparison theorems", or the "cut locus" to minimal surfaces? 17. By considering the Dirac operator on auxiliary vector bundles over compact manifolds, Gromov and Lawson [GLl] extended Lichnerowicz's positive scalar curvature obstructions to a much larger class of manifolds. Can one also use harmonic 1-forms on auxiliary vector bundles to extend Bochner's classical obstructions to positive Ricci curvature? 1 8 . (a) In the recent work of Brezis and Coron [BC] on the Rellich problem on surfaces with constant mean curvature, they only consider maps from the unit disc. What can one say if one seeks surfaces of other topological type and removes the assumption that the mean curvature is a constant? (b) There are two spherical caps in R3 that span a circle and have constant mean curvature H < 1. These are the two obvious solutions of the Rellich problem. Are these the only solutions? 19. Minimal graphs of codimension one are fairly well understood. Lawson and Osserman [LO] have found that for higher codimension new phenomena arise concerning the homotopy class of the boundary values. One should understand this much better. Surely these same issues arise for the Dirichlet problem for more general quasilinear elliptic systems, but nothing is known.

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